Properties

Label 22.2.c.a
Level 22
Weight 2
Character orbit 22.c
Analytic conductor 0.176
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.175670884447\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{10} q^{2} + ( -1 + \zeta_{10} - \zeta_{10}^{3} ) q^{3} + \zeta_{10}^{2} q^{4} + ( -2 + 2 \zeta_{10}^{3} ) q^{5} + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} -2 \zeta_{10}^{2} q^{7} -\zeta_{10}^{3} q^{8} + ( 3 - 2 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{9} +O(q^{10})\) \( q -\zeta_{10} q^{2} + ( -1 + \zeta_{10} - \zeta_{10}^{3} ) q^{3} + \zeta_{10}^{2} q^{4} + ( -2 + 2 \zeta_{10}^{3} ) q^{5} + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} -2 \zeta_{10}^{2} q^{7} -\zeta_{10}^{3} q^{8} + ( 3 - 2 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{9} + ( 2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{10} + ( 2 - 4 \zeta_{10} + 2 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{11} + ( 1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{12} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{13} + 2 \zeta_{10}^{3} q^{14} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{15} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{16} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{17} + ( -3 \zeta_{10} + 2 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{18} + ( -3 + 3 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{19} + ( -2 - 2 \zeta_{10}^{2} ) q^{20} + ( -2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{21} + ( -3 + \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{22} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{23} + ( 1 - 2 \zeta_{10} + \zeta_{10}^{2} ) q^{24} + ( 4 - 4 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{25} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{26} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{28} + ( 4 \zeta_{10} - 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{29} + ( 2 - 2 \zeta_{10} ) q^{30} -2 \zeta_{10} q^{31} + q^{32} + ( -1 + 4 \zeta_{10} - 7 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{33} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{34} + ( 4 + 4 \zeta_{10}^{2} ) q^{35} + ( -3 + 3 \zeta_{10} + \zeta_{10}^{3} ) q^{36} + ( -6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{37} + ( 4 - \zeta_{10} + \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{38} + ( 2 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{39} + ( 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{40} + ( 1 - \zeta_{10} - 5 \zeta_{10}^{3} ) q^{41} + ( -2 + 4 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{42} + ( -3 - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{43} + ( 1 + 2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{44} + ( -8 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{45} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{46} + ( -4 + 4 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{47} + ( -\zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{48} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{49} + ( -3 - \zeta_{10} + \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{50} + ( -2 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{51} + ( 2 - 2 \zeta_{10} ) q^{52} + ( -4 + 8 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{53} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{54} + ( 6 \zeta_{10} + 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{55} -2 q^{56} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{57} + ( 4 - 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{58} + ( -\zeta_{10} - 7 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{59} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{60} + ( 4 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{61} + 2 \zeta_{10}^{2} q^{62} + ( 6 - 6 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{63} -\zeta_{10} q^{64} + 4 q^{65} + ( 4 - 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{66} + ( 8 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{67} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{68} + ( 4 - 4 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{69} + ( -4 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{70} + ( 4 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{71} + ( 1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{72} + ( -\zeta_{10} + 12 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{73} + ( -6 + 6 \zeta_{10} ) q^{74} + ( -5 + 6 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{75} + ( -4 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{76} + ( -2 - 4 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{77} + ( -2 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{78} + ( -12 + 6 \zeta_{10} - 12 \zeta_{10}^{2} ) q^{79} + ( 2 - 2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{80} + ( 6 \zeta_{10} - 2 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{81} + ( -5 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{82} + ( -5 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{83} + ( 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{84} + 2 \zeta_{10}^{3} q^{85} + ( 9 - 6 \zeta_{10} + 9 \zeta_{10}^{2} ) q^{86} + ( -2 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{87} + ( -2 + \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{88} + ( -5 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{89} + ( 2 + 6 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{90} + ( -4 + 4 \zeta_{10} ) q^{91} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{92} + ( -2 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{93} + ( 4 - 4 \zeta_{10}^{3} ) q^{94} + ( -8 \zeta_{10} - 6 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{95} + ( -1 + \zeta_{10} - \zeta_{10}^{3} ) q^{96} + ( 3 - 12 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{97} -3 q^{98} + ( 3 - 4 \zeta_{10} + 8 \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} - 4q^{3} - q^{4} - 6q^{5} + q^{6} + 2q^{7} - q^{8} + 7q^{9} + O(q^{10}) \) \( 4q - q^{2} - 4q^{3} - q^{4} - 6q^{5} + q^{6} + 2q^{7} - q^{8} + 7q^{9} + 4q^{10} - q^{11} + 6q^{12} - 4q^{13} + 2q^{14} + 6q^{15} - q^{16} + 2q^{17} - 8q^{18} - 5q^{19} - 6q^{20} - 12q^{21} - 11q^{22} - 4q^{23} + q^{24} + 9q^{25} + 6q^{26} + 5q^{27} + 2q^{28} + 10q^{29} + 6q^{30} - 2q^{31} + 4q^{32} + 11q^{33} + 2q^{34} + 12q^{35} - 8q^{36} - 18q^{37} + 10q^{38} - 6q^{39} + 4q^{40} - 2q^{41} - 2q^{42} + 6q^{43} + 4q^{44} - 28q^{45} - 4q^{46} - 8q^{47} - 4q^{48} + 3q^{49} - 11q^{50} - 7q^{51} + 6q^{52} - 4q^{53} + 4q^{55} - 8q^{56} + 5q^{57} + 10q^{58} + 5q^{59} - 4q^{60} + 8q^{61} - 2q^{62} + 16q^{63} - q^{64} + 16q^{65} + 16q^{66} + 22q^{67} + 2q^{68} + 14q^{69} - 8q^{70} + 8q^{71} + 7q^{72} - 14q^{73} - 18q^{74} - 9q^{75} - 10q^{76} - 8q^{77} - 16q^{78} - 30q^{79} + 4q^{80} + 14q^{81} - 7q^{82} - 19q^{83} + 8q^{84} + 2q^{85} + 21q^{86} - 20q^{87} - q^{88} - 10q^{89} + 12q^{90} - 12q^{91} + 6q^{92} + 2q^{93} + 12q^{94} - 10q^{95} - 4q^{96} - 3q^{97} - 12q^{98} - 13q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(\zeta_{10}^{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.809017 + 0.587785i 0.118034 + 0.363271i 0.309017 0.951057i −2.61803 1.90211i −0.309017 0.224514i −0.618034 + 1.90211i 0.309017 + 0.951057i 2.30902 1.67760i 3.23607
5.1 0.309017 + 0.951057i −2.11803 1.53884i −0.809017 + 0.587785i −0.381966 + 1.17557i 0.809017 2.48990i 1.61803 1.17557i −0.809017 0.587785i 1.19098 + 3.66547i −1.23607
9.1 0.309017 0.951057i −2.11803 + 1.53884i −0.809017 0.587785i −0.381966 1.17557i 0.809017 + 2.48990i 1.61803 + 1.17557i −0.809017 + 0.587785i 1.19098 3.66547i −1.23607
15.1 −0.809017 0.587785i 0.118034 0.363271i 0.309017 + 0.951057i −2.61803 + 1.90211i −0.309017 + 0.224514i −0.618034 1.90211i 0.309017 0.951057i 2.30902 + 1.67760i 3.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.2.c.a 4
3.b odd 2 1 198.2.f.e 4
4.b odd 2 1 176.2.m.c 4
5.b even 2 1 550.2.h.h 4
5.c odd 4 2 550.2.ba.c 8
8.b even 2 1 704.2.m.h 4
8.d odd 2 1 704.2.m.a 4
11.b odd 2 1 242.2.c.c 4
11.c even 5 1 inner 22.2.c.a 4
11.c even 5 1 242.2.a.f 2
11.c even 5 2 242.2.c.a 4
11.d odd 10 1 242.2.a.d 2
11.d odd 10 1 242.2.c.c 4
11.d odd 10 2 242.2.c.d 4
33.f even 10 1 2178.2.a.x 2
33.h odd 10 1 198.2.f.e 4
33.h odd 10 1 2178.2.a.p 2
44.g even 10 1 1936.2.a.n 2
44.h odd 10 1 176.2.m.c 4
44.h odd 10 1 1936.2.a.o 2
55.h odd 10 1 6050.2.a.ci 2
55.j even 10 1 550.2.h.h 4
55.j even 10 1 6050.2.a.bs 2
55.k odd 20 2 550.2.ba.c 8
88.k even 10 1 7744.2.a.cy 2
88.l odd 10 1 704.2.m.a 4
88.l odd 10 1 7744.2.a.cz 2
88.o even 10 1 704.2.m.h 4
88.o even 10 1 7744.2.a.bm 2
88.p odd 10 1 7744.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.2.c.a 4 1.a even 1 1 trivial
22.2.c.a 4 11.c even 5 1 inner
176.2.m.c 4 4.b odd 2 1
176.2.m.c 4 44.h odd 10 1
198.2.f.e 4 3.b odd 2 1
198.2.f.e 4 33.h odd 10 1
242.2.a.d 2 11.d odd 10 1
242.2.a.f 2 11.c even 5 1
242.2.c.a 4 11.c even 5 2
242.2.c.c 4 11.b odd 2 1
242.2.c.c 4 11.d odd 10 1
242.2.c.d 4 11.d odd 10 2
550.2.h.h 4 5.b even 2 1
550.2.h.h 4 55.j even 10 1
550.2.ba.c 8 5.c odd 4 2
550.2.ba.c 8 55.k odd 20 2
704.2.m.a 4 8.d odd 2 1
704.2.m.a 4 88.l odd 10 1
704.2.m.h 4 8.b even 2 1
704.2.m.h 4 88.o even 10 1
1936.2.a.n 2 44.g even 10 1
1936.2.a.o 2 44.h odd 10 1
2178.2.a.p 2 33.h odd 10 1
2178.2.a.x 2 33.f even 10 1
6050.2.a.bs 2 55.j even 10 1
6050.2.a.ci 2 55.h odd 10 1
7744.2.a.bm 2 88.o even 10 1
7744.2.a.bn 2 88.p odd 10 1
7744.2.a.cy 2 88.k even 10 1
7744.2.a.cz 2 88.l odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(22, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( 1 + 4 T + 3 T^{2} - 10 T^{3} - 29 T^{4} - 30 T^{5} + 27 T^{6} + 108 T^{7} + 81 T^{8} \)
$5$ \( 1 + 6 T + 11 T^{2} + 6 T^{3} + T^{4} + 30 T^{5} + 275 T^{6} + 750 T^{7} + 625 T^{8} \)
$7$ \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 140 T^{5} - 147 T^{6} - 686 T^{7} + 2401 T^{8} \)
$11$ \( 1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4} \)
$13$ \( 1 + 4 T + 3 T^{2} + 50 T^{3} + 341 T^{4} + 650 T^{5} + 507 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 340 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 2755 T^{5} + 7581 T^{6} + 34295 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 + 2 T + 42 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 5800 T^{5} + 26071 T^{6} - 243890 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 2 T - 27 T^{2} - 116 T^{3} + 605 T^{4} - 3596 T^{5} - 25947 T^{6} + 59582 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 18 T + 107 T^{2} + 210 T^{3} + T^{4} + 7770 T^{5} + 146483 T^{6} + 911754 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 2 T - 17 T^{2} - 236 T^{3} + 525 T^{4} - 9676 T^{5} - 28577 T^{6} + 137842 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 - 3 T - 13 T^{2} - 129 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 17860 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 4 T + 43 T^{2} + 380 T^{3} + 4761 T^{4} + 20140 T^{5} + 120787 T^{6} + 595508 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 5 T + T^{2} + 335 T^{3} - 1164 T^{4} + 19765 T^{5} + 3481 T^{6} - 1026895 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 8 T + 3 T^{2} - 436 T^{3} + 6905 T^{4} - 26596 T^{5} + 11163 T^{6} - 1815848 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 - 11 T + 133 T^{2} - 737 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 - 8 T - 47 T^{2} + 434 T^{3} + 1365 T^{4} + 30814 T^{5} - 236927 T^{6} - 2863288 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 14 T + 63 T^{2} - 500 T^{3} - 10579 T^{4} - 36500 T^{5} + 335727 T^{6} + 5446238 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 30 T + 461 T^{2} + 5400 T^{3} + 52861 T^{4} + 426600 T^{5} + 2877101 T^{6} + 14791170 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 19 T + 103 T^{2} + 695 T^{3} + 10536 T^{4} + 57685 T^{5} + 709567 T^{6} + 10863953 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 5 T + 153 T^{2} + 445 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + 3 T + 47 T^{2} - 255 T^{3} + 3496 T^{4} - 24735 T^{5} + 442223 T^{6} + 2738019 T^{7} + 88529281 T^{8} \)
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